I recently read this book for the fourth time and as before, came away amazed at the insights that surfaced. Sharpe's book is by far, one of the most beautiful books on Differential Geometry I have come across. For more than a decade and half I have been enamored of Differential Geometry and Topology and have pursued the beauties of this discipline along the waters of the great deep of Theoretical Physics. I began my journey in this fascinating domain by working through a wholly unconventional monograph. While yet a student of engineering, I stumbled across T J Wilmore's monograph 'Total Curvature in Riemannian Geometry.' Hardly a book for beginners let alone others, Wilmore's book was difficult to say the least. yet it was utterly fascinating. It opened up the road to a more serious and systematic study of Differential Geometry. Wilmore introduced me to the monographs of Sigurdur Helgason and Kobayashi and Nomizu. I rushed headlong into these books and bounced away due to the dense mass of material that one needed to predigest before these books' pronouncements became intelligible. Sobering down, I took on the milder Auslander and Mckenzie's 'Introduction to Differentiable manifolds', Bishop and Crittenden's 'Geometry of manifolds', Milnor's 'Topology from the Differentiable Viewpoint'.
But the capital period of my understanding of the domain came after reading Arnold's 'Mathematical Methods of Classical Mechanics', Abharam's Foundations of Mechanics (the first edition of the now famous Abharam-Marsden). But it was only after encountering the remarkable book of Robert Hermann, 'Differential Geometry and the Calculus of Variations' that the full appreciation of Elie Cartan's methods came home to me. I was already familiar with Cartan after reading about him in Misner, Throne and Wheeler's 'Gravitation'. In that book there is a brief biographical sketch on Cartan that is at once captivating to one getting interested in Manifolds, Lie Groups and Topology. MTW quote from Chern and Chevelley's obituary, 'Eli Cartan and His Mathematical Work'. ( now available for open access at http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183516693).
I tried desperately to get hold of that article but to no avail. Disappointed, I dug into all available sources dealing with Cartan's work. I pursued Dieudonne's treatise on Analysis, Walter Thirring's Course on Mathematical Physics for applications. Later, I found an extensive application of Cartan's methods in Kastrup's 'Canonical theories of Lagrangian dynamical systems in physics' and Friedrich W. Hehla, J. Dermott McCreab, Eckehard W. Mielkea and Yuval Ne'eman's, Metric Affine Gauge theory of Gravity. It was during this time that again, accidentally, Sharpe's book came to me as a heavenly gift. The very first pages struck me and I knew this was the book I had been waiting for all along to take my appreciation of Cartan to completion, if one could speak of such a thing of an ever evolving insight.
Reading it once again made me realize in full the colossal magnitude of the work of Cartan that Chern and Chevelley speak of. You read Cartan for the first time and don't understand. You read a second time and still you don't. You read for the third time and only then you begun to faintly perceive that something profound is being said. More precisely, in Robert Bryant's words, 'You read the introduction to a paper of Cartan and you understand nothing. Then you read the rest of the paper and still you understand nothing. Then you go back and read the introduction again and there begins to be the faint glimmer of something very interesting.' Fortunately, Sharpe's book began to yield to me right after the second reading. The third was enlightening and the fourth sheer enjoyment. It is then that I decided to set forth my appreciation of Cartan and Klein in the present blog. And I choose to begin with a review of Sharpe's book starting here and in my next post.